Ending and consequences of Terry Tao’s criticism



Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

Comparison with lattice dataAs generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

Back to work


Today I am back in my office and physics, that I never stopped to think of, is unfriendly urging in my mind. Besides my work as a reviewer for Mathematical Reviews that I try to do at my best, I have not forgotten to look around in the blogosphere. In these days there is a lot of fuss about a recent paper by Fermi Collaboration (see here). You can find some discussion here but it is not the only blog discussing this matter: An inflamed discussion is also here. Is loop quantum gravity dead? I can only spend a few words here by saying that is really too early to draw such conclusions but the results from Fermi Collaboration are really beautiful and open the premises for a bright future in the observation of gamma ray bursts. I would like to remember here the point of view of Steven Weinberg claiming that we finally hit exact truths on Nature (the bitch not the journal as  Tommaso Dorigo uses to say ). These truths are special relativity and quantum mechanics. I share this idea and the more beautiful idea that we are indeed able to reach such exact truths, the same mathematicians are able to achieve. On the same ground, if some experiment should come out claiming that these theories should be modified, I would be glad for one thing: A great opportunity for my generation to put hands on a deeper truth.

I am still working on a perturbation analysis of a non-perturbative Higgs field interacting with a fermion. I am solving Heisenberg equations of motion with a very large coupling for the Higgs. I hope to get some further time to complete the computations that may become really involved.

About QCD there is few to say. I have got my paper accepted for publication as you may know that implied a nice correspondence with Terry Tao. We can say that, after the very good Terry’s intervention, the proof of the mapping theorem is indeed complete. Another paper hit my interest and arose from a Japanese group working on lattice (see here). I have had an email exachange with Hideo Suganuma and surely their results are something to think about in depth. I hope to see other papers in the near future as this area of physics is very active indeed.

Finally, great expectations are for LHC. It will start on November and we hope to see results very soon. Infancy problems should be overcome and it is time to see the face of Higgs (the particle not the professor). Good luck, folks!

Answer to Terry Tao’s criticism will go published


My paper containing the answer to Terry Tao’s criticism will be published in Modern Physics Letters A. You can get a copy of this preprint from arxiv here.

Thank you very much, folks!



Today I have posted a paper on arxiv. It will appear on monday. This paper was required to my by Terry Tao to supplement the proof of the mapping theorem showing that indeed it holds.

If you cannot hold the paper is 09032357v1: preprint. Don’t trust that number as may change.

The argument may be put up with very simple words: If you trust Smilga’s solutions that depend only on time, a Lorentz boost will fit the bill.

I did it for you


It is very easy to show, from Yang-Mills equations, how to obtain a scalar field equation through the Smilga’s choice. Let us write down Yang-Mills equations

\partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

using the choice A_1^1=A_2^2=A_3^3=\phi. This is really a great simplification. Smilga, in his book, already checked this for us but we give here the full computation. From above eqautions, the only critical term is the following

f^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

as this term would produce terms deviating from the known form of the scalar theory. For SU(2) we have f^{abc}=\epsilon^{abc} the fully-antisymmetric Levi-Civita tensor. This means that we will have

\epsilon^{a1c}A^{11}(\partial_1A_\nu^c-\partial_\nu A_1^c)+

\epsilon^{a2c}A^{22}(\partial_2A_\nu^c-\partial_\nu A_2^c)+

\epsilon^{a3c}A^{33}(\partial_3A_\nu^c-\partial_\nu A_3^c).

Where we have used largely Smilga’s choice. Now do the following. Take the following components to evolve \nu=1 a=1, \nu=2  a=2 and \nu=3, a=3. It easy to see that the possible harmful term is zero with the Smilgaì’s choice. Now, for the cubic term you should use the useful relation


and you will get back the quartic term.

The gauge fixing term can be easily disposed of through a rescaling of spatial variables while the kinematic term gives the right contribution. You will get three identical equations for the scalar field.

Of course, Smilga in his book already did this and I repeated his computations after the Editor of PLB asked for a revision having the referee already put out this problem. The Editorial work was done very well and two referees read the paper emphasizing errors where they were.

Finally, Tao’s critcism does not apply as I said. This does not mean that what he says is wrong. This means that does not apply to my case.

Update: As the question of the gauge fixing term appears so relevant, let me fix it once and for all. Firstly, I would like to point out that these solutions belong to a class of solutions in the Maximal Abelian Gauge (MAG). But let us forget about this and consider the question of gauge fixing. This term is arbitrarily introduced in the Lagrangian of the field in order to fix the gauge when a quantization procedure is applied. Due to gauge invariance and the fact that becomes an exact differential after partial integration, it useful to have it there for the above aims. The form that it  takes is

\frac{1}{\alpha}(\partial A)^2

and is put directly into the Lagrangian. How does this term become with the Smilga’s choice? One has


and the final effect is a pure rescaling into the space variables of the scalar field. In this way the argument is made consistent. One cannot take the other way around for the very nature of this term and claiming the result is wrong.

This particular class of solutions belongs to the subgroup of SU(N) given by the direct product of U(1). This is a property of MAG and all the matter is really consistent and works.

Finally, I invite people commenting this and other posts to limit herself to polite responses and in the realm of scientific discussion. Of course, doing something wrong happens and happened to anyone working in a scientifc endeavour for the simple reason that she is really doing things. People that only do useless criticisms boiling down to personal offenses are kindly invited to refrain from further interventions.

Update 2: I will get a paper published about this matter. Please, check here.

Is Terry wrong?


I am a great estimator of Terry Tao and a reader of his blog. Tao is a Fields medalist and one of the greatest living mathematicians. Relying on such a giant authority may give someone the feeling of being a kind of dwarf trying to be listened around. Anyhow I will try. Terry come out with an intervention in Wikipedia here claiming:

“It may be relevant to point out that one of the references cited in the disputed section [3] has a significant error in it, despite being published. Namely, in the proof of Theorem 1, the author is assuming that an extremum A for the Yang-Mills action for a special class of connections (namely those in which A^1_1=A^2_2=A^3_3 and all other components vanish) is necessarily an extremum for the Yang-Mills action for all other connections also, but this is not the case (just because YM(A) \geq YM(A'), for instance, for A’ of this special form, does not imply that YM(A) \geq YM(A') for general A’). Since one needs to be an extremiser (or critical point) in the space of all connections in order to be a solution to the Yang-Mills equations, the mapping provided in Theorem 1 has not been shown to actually produce solutions to the Yang-Mills equation (and I suspect that if one actually checks the Yang-Mills equation for this mapping, that one will not in fact get such a solution). Terry (talk) 20:32, 28 February 2009 (UTC)”

This claim of mistake by my side contains a misinterpretation of the mapping theorem. If the theorem would claim that this is true for all connections, as Terry says, it would be istantaneously false. I cannot map a scalar field on all the Y-M connections (think of chaotic solutions). The theorem simply states that there exists a class of solutions of the quartic scalar field that are also solution for the Yang-Mills equations and this can be easily proved by substitution (check Smilga’s book) and Tao is proved istantaneously wrong. So now, what is the point? I have a class of Yang-Mills solutions that Tao is claiming are not. But whoever can check by herself that I am right. So, is Terry wrong?


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